Nilpotent Lie Groups: Structure and Applications to Analysis
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562 Roe W. Goodman
Nilpotent Lie Groups:
Structure and Applications to Analysis
Springer-Verlag Berlin-Heidelberg • New York 1976
Author Roe William Goodman Department of Mathematics Rutgers The State University New Brunswick, N. J. 0 8 9 0 3 / U S A
Library of Congress Cataloging in Publication Data
Goodman, Roe. Nilpotent lie groups. (Lecture notes in mathematics ; 562) Bibliography: p. Includes index. i. Lie groups, Nilpotent. 2o Representetions of groups. 3° Differential equations~ Hypoelliptic. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 562. QA3. L28 no. 562 [QA387 ] 512'.55 76-30271
AMS Subject Classifications (1970): 44A25, 17B30, 22E25, 22E30, 22E45, 35H05, 32M15 ISBN 3-540-08055-4 Springer-Verlag Berlin • Heidelberg ' New York ISBN 0-38?-08055-4 Springer-Verlag New York • Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
Table o f Contents
Chapter I ,
Structure of nilpotent
L i e algebras and L i e groups . . . . . . . . . . . . . .
§ 1. D e r i v a t i o n s and automorphisms o f f i l t e r e d I.I
1
polynomial r i n g s
D i l a t i o n s and g r a d a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I
1,2 Homogeneous norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3 Vector f i e l d s
4
w i t h polynomial c o e f f i c i e n t s . . . . . . . . . . . . . . . . . . . . . . . .
1.4 L o c a l l y u n i p o t e n t automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.5 Transformation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i0
1.6 F i n i t e dimensional r e p r e s e n t a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I0
1.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
II
§ 2. B i r k h o f f embedding theorem 2,1 F i l t r a t i o n s
on n i l p o t e n t
Lie a l g e b r a s . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.2 A l g e b r a i c comparison o f a d d i t i v e and n i l p o t e n t group s t r u c t u r e s . . 13 2.3 F a i t h f u l
unipotent representations ...............................
16
§ 3. Comparison o f group s t r u c t u r e s 3.1 Norm comparison o f a d d i t i v e and n i l p o t e n t 3.2 A l g e b r a i c comparison o f f i l t e r e d 3.3 Norm comparison o f
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